According to the Bloch--Beilinson conjectures, an automorphism of a K3
surface $X$ that acts as the identity on the transcendental lattice should
act trivially on $\CH^2(X)$. We discuss this conjecture for symplectic
involutions and prove it in one third of all cases. The main point is
to use special elliptic K3 surfaces and stable maps to produce covering
families of elliptic curves on the generic K3 surface that are invariant
under the involution.